Space and Lobatchewsky's Geometrical Researches on the Theory of Parallels (1840) were rendered easily accessible to American readers by translations into English made in 1891 by George Bruce Halsted of the University of Texas.
The Russian and Hungarian mathematicians were not the only ones to whom pangeometry suggested itself. A copy of the Tentamen reached Gauss, the elder Bolyai's former room-mate at Göttingen, and this Nestor of German mathematicians was surprised to discover in it worked out what he himself had begun long before, only to leave it after him in his papers. As early as 1792 he had started on researches of that character. His letters show that in 1799 he was trying to prove a priori the reality of Euclid's system; but some time within the next thirty years he arrived at the conclusion reached by Lobatchewsky and Bolyai. In 1829 he wrote to Bessel, stating that his "conviction that we cannot found geometry completely a priori has become, if possible, still firmer," and that "if number is merely a product of our mind, space has also a reality beyond our mind of which we cannot fully foreordain the laws a priori." The term non-Euclidean geometry is due to Gauss. It has recently been brought to notice that Geronimo Saccheri, a Jesuit father of Milan, in 1733 anticipated Lobatchewsky's doctrine of the parallel angle. Moreover, G. B. Halsted has pointed out that in 1766 Lambert wrote a paper "Zur Theorie der Parallellinien," published in the Leipziger Magazin für reine und angewandte Mathematik, 1786, in which: (1) The failure of the parallel-axiom in surface-spherics gives a geometry with angle-sum > 2 right angles; (2) In order to make intuitive a geometry with angle-sum < 2 right angles we need the aid of an "imaginary sphere" (pseudo-sphere); (3) In a space with the angle-sum differing from 2 right angles, there is an absolute measure (Bolyai's natural unit for length).
